Verfügbarkeit: Quantum computing

Quantum computing: from linear algebra to physical realizations

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Hauptverfasser:	Nakahara, Mikio (VerfasserIn), Ohmi, Tetsuo (VerfasserIn)
Format:	Buch
Sprache:	English
Veröffentlicht:	Boca Raton, FL. [u.a.] CRC Press 2008
Schlagworte:	Quantencomputer
Online-Zugang:	Inhaltsverzeichnis
Beschreibung:	XVI, 421 S.
ISBN:	0750309830 9780750309837

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adam_text	Contents I From Linear Algebra to Quantum Computing 1 1 Basics of Vectors and Matrices 3 1.1 Vector Spaces ........................... 4 1.2 Linear Dependence and Independence of Vectors ....... 5 1.3 Dual Vector Spaces ........................ 6 1.4 Basis, Projection Operator and Completeness Relation .... 8 1.4.1 Orthonormal Basis and Completeness Relation .... 8 1.4.2 Projection Operators ................... 9 1.4.3 Gram-Schmidt Orthonormalization ........... 10 1.5 Linear Operators and Matrices ................. 11 1.5.1 Hermitian Conjugate, Hermitian and Unitary Matrices .......................... 12 1.6 Eigenvalue Problems ....................... 13 1.6.1 Eigenvalue Problems of Hermitian and Normal Matrices .......................... 14 1.7 Pauli Matrices .......................... 18 1.8 Spectral Decomposition ..................... 19 1.9 Singular Value Decomposition (SVD) .............. 23 1.10 Tensor Product (Kronecker Product) .............. 26 2 Framework of Quantum Mechanics 29 2.1 Fundamental Postulates ..................... 29 2.2 Some Examples .......................... 32 2.3 Multipartite System, Tensor Product and Entangled State . . 36 2.4 Mixed States and Density Matrices ............... 38 2.4.1 Negativity ......................... 42 2.4.2 Partial Trace and Purification .............. 45 2.4.3 Fidelity .......................... 47 3 Qubits and Quantum Key Distribution 51 3.1 Qubits ............................... 51 3.1.1 One Qubit ......................... 51 3.1.2 Bloch Sphere ....................... 53 3.1.3 Multi-Qubit Systems and Entangled States ...... 54 3.1.4 Measurements ....................... 56 3.1.5 Einstein-Podolsky- Rosen (EPR) Paradox ....... 59 3.2 Quantum Key Distribution (BB84 Protocol) .......... 60 4 Quantum Gates, Quantum Circuit and Quantum Computa¬ tion 65 4.1 Introduction............................ 65 4.2 Quantum Gates.......................... 66 4.2.1 Simple Quantum Gates ................. 66 4.2.2 Walsh-Hadamard Transformation............ 69 4.2.3 SWAP Gate and Fredkin Gate............. 70 4.3 Correspondence with Classical Logic Gates .......... 71 4.3.1 NOT Gate......................... 72 4.3.2 XOR Gate......................... 72 4.3.3 AND Gate......................... 73 4.3.4 OR Gate.......................... 73 4.4 No-Cloning Theorem....................... 75 4.5 Dense Coding and Quantum Teleportation ........... 76 4.5.1 Dense Coding ....................... 77 4.5.2 Quantum Teleportation ................. 79 4.6 Universal Quantum Gates .................... 82 4.7 Quantum Parallelism and Entanglement ............ 95 5 Simple Quantum Algorithms 99 5.1 Deutsch Algorithm ........................ 99 5.2 Deutsch-Jozsa Algorithm and Bernstein-Vazirani Algorithm . 101 5.3 Simon s Algorithm ........................ 105 6 Quantum Integral Transforms 109 6.1 Quantum Integral Transforms .................. 109 6.2 Quantum Fourier Transform (QFT) .............. Ill 6.3 Application of QFT: Period-Finding .............. 113 6.4 Implementation of QFT ..................... 116 6.5 Walsh-Hadamard Transform ................... 122 6.6 Selective Phase Rotation Transform .............. 123 7 Grover s Search Algorithm 125 7.1 Searching for a Single File .................... 125 7.2 Searching for d Files ....................... 133 8 Shor s Factorization Algorithm 137 8.1 The RSA Cryptosystem ..................... 137 8.2 Factorization Algorithm ..................... 140 8.3 Quantum Part of Shor s Algorithm ............... 141 8.3.1 Settings for STEP 2................... 141 8.3.2 STEP 2.......................... 143 8.4 Probability Distribution ..................... 144 8.5 Continued Fractions and Order Finding ............ 151 8.6 Modular Exponential Function ................. 156 8.6.1 Adder ........................... 157 8.6.2 Modular Adder ...................... 161 8.6.3 Modular Multiplexer ................... 166 8.6.4 Modular Exponential Function ............. 168 8.6.5 Computational Complexity of Modular Exponential Circuit ........................... 170 9 Decoherence 173 9.1 Open Quantum System ..................... 173 9.1.1 Quantum Operations and Kraus Operators ...... 174 9.1.2 Operator-Sum Representation and Noisy Quantum Channel .......................... 177 9.1.3 Completely Positive Maps ................ 178 9.2 Measurements as Quantum Operations ............. 179 9.2.1 Projective Measurements ................ 179 9.2.2 POVM ........................... 180 9.3 Examples ............................. 181 9.3.1 Bit-Flip Channel ..................... 181 9.3.2 Phase-Flip Channel .................... 183 9.3.3 Depolarizing Channel .................. 185 9.3.4 Amplitude-Damping Channel .............. 187 9.4 Lindblad Equation ........................ 188 9.4.1 Quantum Dynamical Semigroup ............ 189 9.4.2 Lindblad Equation .................... 189 9.4.3 Examples ......................... 192 10 Quantum Error Correcting Codes 195 10.1 Introduction ............................195 10.2 Three-Qubit Bit-Flip Code and Phase-Flip Code .......196 10.2.1 Bit-Flip QECC ......................196 10.2.2 Phase-Flip QECC ....................202 10.3 Shor s Nine-Qubit Code .....................203 10.3.1 Encoding .........................204 10.3.2 Transmission .......................205 10.3.3 Error Syndrome Detection and Correction .......205 10.3.4 Decoding .........................208 10.4 Seven-Qubit QECC ........................209 10.4.1 Classical Theory of Error Correcting Codes ......209 10.4.2 Seven-Qubit QECC ....................213 10.4.3 Gate Operations for Seven-Qubit QECC ........220 10.5 Five-Qubit QECC ........................224 10.5.1 Encoding .........................224 10.5.2 Error Syndrome Detection ................227 II Physical Realizations of Quantum Computing 231 11 Di Vincenzo Criteria 233 11.1 Introduction ............................233 11.2 DiVincenzo Criteria .......................234 11.3 Physical Realizations .......................239 12 NMR Quantum Computer 241 12.1 Introduction ............................241 12.2 NMR Spectrometer ........................241 12.2.1 Molecules .........................242 12.2.2 NMR Spectrometer ....................242 12.3 Hamiltonian ............................245 12.3.1 Single-Spin Hamiltonian .................245 12.3.2 Multi-Spin Hamiltonian .................248 12.4 Implementation of Gates and Algorithms ............252 12.4.1 One-Qubit Gates in One-Qubit Molecule ........252 12.4.2 One-Qubit Operation in Two-Qubit Molecule: Bloch- Siegert Effect .......................256 12.4.3 Two-Qubit Gates .....................257 12.4.4 Multi-Qubit Gates ....................259 12.5 Time-Optimal Control of NMR Quantum Computer .....262 12.5.1 A Brief Introduction to Lie Algebras and Lie Groups . 262 12.5.2 Cartan Decomposition and Optimal Implementation of Two-Qubit Gates ...................264 12.6 Measurements ...........................268 12.6.1 Introduction and Preliminary ..............268 12.6.2 One-Qubit Quantum State Tomography ........269 12.6.3 Free Induction Decay (FID) ...............270 12.6.4 Two-Qubit Tomography .................271 12.7 Preparation of Pseudopure State ................274 12.7.1 Temporal Averaging ...................276 12.7.2 Spatial Averaging .....................277 12.8 DiVincenzo Criteria .......................281 13 Trapped Ions 285 13.1 Introduction ............................285 13.2 Electronic States of Ions as Qubits ...............287 13.3 Ions in Paul Trap .........................289 13.3.1 Trapping Potential ....................289 13.3.2 Lattice Formation ....................294 13.3.3 Normal Modes ......................296 13.4 Ion Qubit .............................298 13.4.1 One-Spin Hamiltonian ..................298 13.4.2 Sideband Cooling .....................301 13.5 Quantum Gates ..........................302 13.5.1 One-Qubit Gates .....................302 13.5.2 CNOT Gate ........................304 13.6 Readout ..............................306 13.7 DiVincenzo Criteria .......................307 14 Quantum Computing with Neutral Atoms 311 14.1 Introduction ............................311 14.2 Trapping Neutral Atoms .....................311 14.2.1 Alkali Metal Atoms .................... 311 14.2.2 Magneto-Optical Trap (MOT) ............. 312 14.2.3 Optical Dipole Trap ................... 314 14.2.4 Optical Lattice ...................... 316 14.2.5 Spin-Dependent Optical Potential ............ 317 14.3 One-Qubit Gates ......................... 319 14.4 Quantum State Engineering of Neutral Atoms ......... 321 14.4.1 Trapping of a Single Atom ................ 321 14.4.2 Rabi Oscillation ...................... 321 14.4.3 Neutral Atom Quantum Regisiter ............ 323 14.5 Preparation of Entangled Neutral Atoms ............ 324 14.6 DiVincenzo Criteria ....................... 327 15 Josephson Junction Qubits 329 15.1 Introduction ............................329 15.2 Nanoscale Josephson Junctions and SQUIDs ..........330 15.2.1 Josephson Junctions ...................330 15.2.2 SQUIDs ..........................333 15.3 Charge Qubit ...........................337 15.3.1 Simple Cooper Pair Box .................337 15.3.2 Split Cooper Pair Box ..................341 15.4 Flux Qubit ............................342 15.4.1 Simplest Flux Qubit ...................342 15.4.2 Three-Junction Flux Qubit ...............345 15.5 Quantronium ...........................347 15.6 Current-Biased Qubit (Phase Qubit) ..............348 15.7 Readout ..............................352 15.7.1 Charge Qubit .......................352 15.7.2 Readout of Quantronium ................355 15.7.3 Switching Current Readout of Flux Qubits ......357 15.8 Coupled Qubits ..........................358 15.8.1 Capacitively Coupled Charge Qubits ..........359 15.8.2 Inductive Coupling of Charge Qubits ..........362 15.8.3 Tunable Coupling between Flux Qubits ........366 15.8.4 Coupling Flux Qubits with an LC Resonator .....369 15.9 DiVincenzo Criteria .......................374 16 Quantum Computing with Quantum Dots 377 16.1 Introduction ............................377 16.2 Mesoscopic Semiconductors ...................377 16.2.1 Two-Dimensional Electron Gas in Inversion Layer . . 377 16.2.2 Coulomb Blockade ....................378 16.3 Electron Charge Qubit ......................383 16.3.1 Electron Charge Qubit ..................384 16.3.2 Rabi Oscillation ......................385 16.4 Electron Spin Qubit .......................386 16.4.1 Electron Spin Qubit ...................386 16.4.2 Single-Qubit Operations .................387 16.4.3 Coherence Time .....................390 16.5 DiVincenzo Criteria .......................396 16.5.1 Charge Qubits ......................396 16.5.2 Spin Qubits ........................397 A Solutions to Selected Exercises 399 Index 417
adam_txt	Contents I From Linear Algebra to Quantum Computing 1 1 Basics of Vectors and Matrices 3 1.1 Vector Spaces . 4 1.2 Linear Dependence and Independence of Vectors . 5 1.3 Dual Vector Spaces . 6 1.4 Basis, Projection Operator and Completeness Relation . 8 1.4.1 Orthonormal Basis and Completeness Relation . 8 1.4.2 Projection Operators . 9 1.4.3 Gram-Schmidt Orthonormalization . 10 1.5 Linear Operators and Matrices . 11 1.5.1 Hermitian Conjugate, Hermitian and Unitary Matrices . 12 1.6 Eigenvalue Problems . 13 1.6.1 Eigenvalue Problems of Hermitian and Normal Matrices . 14 1.7 Pauli Matrices . 18 1.8 Spectral Decomposition . 19 1.9 Singular Value Decomposition (SVD) . 23 1.10 Tensor Product (Kronecker Product) . 26 2 Framework of Quantum Mechanics 29 2.1 Fundamental Postulates . 29 2.2 Some Examples . 32 2.3 Multipartite System, Tensor Product and Entangled State . . 36 2.4 Mixed States and Density Matrices . 38 2.4.1 Negativity . 42 2.4.2 Partial Trace and Purification . 45 2.4.3 Fidelity . 47 3 Qubits and Quantum Key Distribution 51 3.1 Qubits . 51 3.1.1 One Qubit . 51 3.1.2 Bloch Sphere . 53 3.1.3 Multi-Qubit Systems and Entangled States . 54 3.1.4 Measurements . 56 3.1.5 Einstein-Podolsky- Rosen (EPR) Paradox . 59 3.2 Quantum Key Distribution (BB84 Protocol) . 60 4 Quantum Gates, Quantum Circuit and Quantum Computa¬ tion 65 4.1 Introduction. 65 4.2 Quantum Gates. 66 4.2.1 Simple Quantum Gates . 66 4.2.2 Walsh-Hadamard Transformation. 69 4.2.3 SWAP Gate and Fredkin Gate. 70 4.3 Correspondence with Classical Logic Gates . 71 4.3.1 NOT Gate. 72 4.3.2 XOR Gate. 72 4.3.3 AND Gate. 73 4.3.4 OR Gate. 73 4.4 No-Cloning Theorem. 75 4.5 Dense Coding and Quantum Teleportation . 76 4.5.1 Dense Coding . 77 4.5.2 Quantum Teleportation . 79 4.6 Universal Quantum Gates . 82 4.7 Quantum Parallelism and Entanglement . 95 5 Simple Quantum Algorithms 99 5.1 Deutsch Algorithm . 99 5.2 Deutsch-Jozsa Algorithm and Bernstein-Vazirani Algorithm . 101 5.3 Simon's Algorithm . 105 6 Quantum Integral Transforms 109 6.1 Quantum Integral Transforms . 109 6.2 Quantum Fourier Transform (QFT) . Ill 6.3 Application of QFT: Period-Finding . 113 6.4 Implementation of QFT . 116 6.5 Walsh-Hadamard Transform . 122 6.6 Selective Phase Rotation Transform . 123 7 Grover's Search Algorithm 125 7.1 Searching for a Single File . 125 7.2 Searching for d Files . 133 8 Shor's Factorization Algorithm 137 8.1 The RSA Cryptosystem . 137 8.2 Factorization Algorithm . 140 8.3 Quantum Part of Shor's Algorithm . 141 8.3.1 Settings for STEP 2. 141 8.3.2 STEP 2. 143 8.4 Probability Distribution . 144 8.5 Continued Fractions and Order Finding . 151 8.6 Modular Exponential Function . 156 8.6.1 Adder . 157 8.6.2 Modular Adder . 161 8.6.3 Modular Multiplexer . 166 8.6.4 Modular Exponential Function . 168 8.6.5 Computational Complexity of Modular Exponential Circuit . 170 9 Decoherence 173 9.1 Open Quantum System . 173 9.1.1 Quantum Operations and Kraus Operators . 174 9.1.2 Operator-Sum Representation and Noisy Quantum Channel . 177 9.1.3 Completely Positive Maps . 178 9.2 Measurements as Quantum Operations . 179 9.2.1 Projective Measurements . 179 9.2.2 POVM . 180 9.3 Examples . 181 9.3.1 Bit-Flip Channel . 181 9.3.2 Phase-Flip Channel . 183 9.3.3 Depolarizing Channel . 185 9.3.4 Amplitude-Damping Channel . 187 9.4 Lindblad Equation . 188 9.4.1 Quantum Dynamical Semigroup . 189 9.4.2 Lindblad Equation . 189 9.4.3 Examples . 192 10 Quantum Error Correcting Codes 195 10.1 Introduction .195 10.2 Three-Qubit Bit-Flip Code and Phase-Flip Code .196 10.2.1 Bit-Flip QECC .196 10.2.2 Phase-Flip QECC .202 10.3 Shor's Nine-Qubit Code .203 10.3.1 Encoding .204 10.3.2 Transmission .205 10.3.3 Error Syndrome Detection and Correction .205 10.3.4 Decoding .208 10.4 Seven-Qubit QECC .209 10.4.1 Classical Theory of Error Correcting Codes .209 10.4.2 Seven-Qubit QECC .213 10.4.3 Gate Operations for Seven-Qubit QECC .220 10.5 Five-Qubit QECC .224 10.5.1 Encoding .224 10.5.2 Error Syndrome Detection .227 II Physical Realizations of Quantum Computing 231 11 Di Vincenzo Criteria 233 11.1 Introduction .233 11.2 DiVincenzo Criteria .234 11.3 Physical Realizations .239 12 NMR Quantum Computer 241 12.1 Introduction .241 12.2 NMR Spectrometer .241 12.2.1 Molecules .242 12.2.2 NMR Spectrometer .242 12.3 Hamiltonian .245 12.3.1 Single-Spin Hamiltonian .245 12.3.2 Multi-Spin Hamiltonian .248 12.4 Implementation of Gates and Algorithms .252 12.4.1 One-Qubit Gates in One-Qubit Molecule .252 12.4.2 One-Qubit Operation in Two-Qubit Molecule: Bloch- Siegert Effect .256 12.4.3 Two-Qubit Gates .257 12.4.4 Multi-Qubit Gates .259 12.5 Time-Optimal Control of NMR Quantum Computer .262 12.5.1 A Brief Introduction to Lie Algebras and Lie Groups . 262 12.5.2 Cartan Decomposition and Optimal Implementation of Two-Qubit Gates .264 12.6 Measurements .268 12.6.1 Introduction and Preliminary .268 12.6.2 One-Qubit Quantum State Tomography .269 12.6.3 Free Induction Decay (FID) .270 12.6.4 Two-Qubit Tomography .271 12.7 Preparation of Pseudopure State .274 12.7.1 Temporal Averaging .276 12.7.2 Spatial Averaging .277 12.8 DiVincenzo Criteria .281 13 Trapped Ions 285 13.1 Introduction .285 13.2 Electronic States of Ions as Qubits .287 13.3 Ions in Paul Trap .289 13.3.1 Trapping Potential .289 13.3.2 Lattice Formation .294 13.3.3 Normal Modes .296 13.4 Ion Qubit .298 13.4.1 One-Spin Hamiltonian .298 13.4.2 Sideband Cooling .301 13.5 Quantum Gates .302 13.5.1 One-Qubit Gates .302 13.5.2 CNOT Gate .304 13.6 Readout .306 13.7 DiVincenzo Criteria .307 14 Quantum Computing with Neutral Atoms 311 14.1 Introduction .311 14.2 Trapping Neutral Atoms .311 14.2.1 Alkali Metal Atoms . 311 14.2.2 Magneto-Optical Trap (MOT) . 312 14.2.3 Optical Dipole Trap . 314 14.2.4 Optical Lattice . 316 14.2.5 Spin-Dependent Optical Potential . 317 14.3 One-Qubit Gates . 319 14.4 Quantum State Engineering of Neutral Atoms . 321 14.4.1 Trapping of a Single Atom . 321 14.4.2 Rabi Oscillation . 321 14.4.3 Neutral Atom Quantum Regisiter . 323 14.5 Preparation of Entangled Neutral Atoms . 324 14.6 DiVincenzo Criteria . 327 15 Josephson Junction Qubits 329 15.1 Introduction .329 15.2 Nanoscale Josephson Junctions and SQUIDs .330 15.2.1 Josephson Junctions .330 15.2.2 SQUIDs .333 15.3 Charge Qubit .337 15.3.1 Simple Cooper Pair Box .337 15.3.2 Split Cooper Pair Box .341 15.4 Flux Qubit .342 15.4.1 Simplest Flux Qubit .342 15.4.2 Three-Junction Flux Qubit .345 15.5 Quantronium .347 15.6 Current-Biased Qubit (Phase Qubit) .348 15.7 Readout .352 15.7.1 Charge Qubit .352 15.7.2 Readout of Quantronium .355 15.7.3 Switching Current Readout of Flux Qubits .357 15.8 Coupled Qubits .358 15.8.1 Capacitively Coupled Charge Qubits .359 15.8.2 Inductive Coupling of Charge Qubits .362 15.8.3 Tunable Coupling between Flux Qubits .366 15.8.4 Coupling Flux Qubits with an LC Resonator .369 15.9 DiVincenzo Criteria .374 16 Quantum Computing with Quantum Dots 377 16.1 Introduction .377 16.2 Mesoscopic Semiconductors .377 16.2.1 Two-Dimensional Electron Gas in Inversion Layer . . 377 16.2.2 Coulomb Blockade .378 16.3 Electron Charge Qubit .383 16.3.1 Electron Charge Qubit .384 16.3.2 Rabi Oscillation .385 16.4 Electron Spin Qubit .386 16.4.1 Electron Spin Qubit .386 16.4.2 Single-Qubit Operations .387 16.4.3 Coherence Time .390 16.5 DiVincenzo Criteria .396 16.5.1 Charge Qubits .396 16.5.2 Spin Qubits .397 A Solutions to Selected Exercises 399 Index 417
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publisher	CRC Press
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spelling	Nakahara, Mikio Verfasser (DE-588)1028332297 aut Quantum computing from linear algebra to physical realizations Mikio Nakahara ; Tetsuo Ohmi Boca Raton, FL. [u.a.] CRC Press 2008 XVI, 421 S. txt rdacontent n rdamedia nc rdacarrier Quantencomputer (DE-588)4533372-5 gnd rswk-swf Quantencomputer (DE-588)4533372-5 s DE-604 Ohmi, Tetsuo Verfasser aut Digitalisierung UB Bayreuth application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016139077&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis
spellingShingle	Nakahara, Mikio Ohmi, Tetsuo Quantum computing from linear algebra to physical realizations Quantencomputer (DE-588)4533372-5 gnd
subject_GND	(DE-588)4533372-5
title	Quantum computing from linear algebra to physical realizations
title_auth	Quantum computing from linear algebra to physical realizations
title_exact_search	Quantum computing from linear algebra to physical realizations
title_exact_search_txtP	Quantum computing from linear algebra to physical realizations
title_full	Quantum computing from linear algebra to physical realizations Mikio Nakahara ; Tetsuo Ohmi
title_fullStr	Quantum computing from linear algebra to physical realizations Mikio Nakahara ; Tetsuo Ohmi
title_full_unstemmed	Quantum computing from linear algebra to physical realizations Mikio Nakahara ; Tetsuo Ohmi
title_short	Quantum computing
title_sort	quantum computing from linear algebra to physical realizations
title_sub	from linear algebra to physical realizations
topic	Quantencomputer (DE-588)4533372-5 gnd
topic_facet	Quantencomputer
url	http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016139077&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA
work_keys_str_mv	AT nakaharamikio quantumcomputingfromlinearalgebratophysicalrealizations AT ohmitetsuo quantumcomputingfromlinearalgebratophysicalrealizations

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